The Pauli Exclusion Principle — Question 7

Question 7

Pauli-allowed configurations need not be ground configurations

An independent-electron model has three spatial orbitals with energies $0$, $\epsilon$, and $3\epsilon$. Each allows two opposite-spin electrons. Four electrons are present; interactions are neglected.

1. Find the lowest-energy occupancy and total energy.

2. Determine whether occupancies $(1,2,1)$ and $(3,1,0)$ are allowed, and find the energy of any allowed listed configuration.

Question 7 - Solution

Pauli-allowed configurations need not be ground configurations

(a) Method and calculation

$$n_1+n_2+n_3=4,\quad 0\le n_i\le2.$$$$\boxed{(n_1,n_2,n_3)=(2,2,0),\quad E_g=2\epsilon}.$$

(b) Interpretation and further calculation

$$(1,2,1):\quad\boxed{\text{allowed},\quad E=2\epsilon+3\epsilon=5\epsilon}.$$$$(3,1,0):\quad n_1=3>2\Rightarrow\boxed{\text{forbidden}}.$$

Physical check

Exclusion constrains allowed states; energy minimization selects a ground state among them. An allowed excitation does not violate Pauli merely because a lower orbital has a vacancy.